Acceleration a is constant.

We know-

d²s /dt² =dv/dt =a 

Integrating a with respect to time will leave us with an expression for velocity, v , (as acceleration is the rate of change of velocity, wrt time):

v=ds/dt = ∫a dt = at+u,

v= u+at  .......(1)

 

where u is a constant of integration. In context it is the initial velocity a body is travelling before it accelerates.

Integrating velocity wrt time will leave us with an expression for displacement (as velocity is the first derivative of displacement modelled as a function of time):

s =∫(ds/dt) dt = ∫(at+u) dt = ∫at dt + ∫u dt

Hence,  s= ut+(1/2)at²                           ...........(2)

Note that this equation assumes displacemnt is zero when time t= 0, i.e. the constant of integration has been neglected.

From equation (2)

s = 1/2(2u+at) t

s= 1/2(u+u+at)t  = 1/2(u+v)t                     

v² = (u+at)² =u² +2uat+at²

v² = u² +2a(ut+1/2 at²)

= u²+2as

v²=u²+2as .............(3)